I battled with this very same question. I’m presently developing my own ability to develop WordPress sites. My employer is launching numerous websites on the engine, and is moving me from general PHP work to focused WordPress development. The thing is, these sites will all be managed by other people, and not myself.
Using widgets allows the non-programmers to help develop the site without potential of breaking things. For instance, they can move things around in the sidebar, change up the contents of the footer, and so much more. That being said, I wish I could prevent them from doing some types of editing, but permit them to do others. While they can’t break the site (to my knowledge), they could really make it ugly.